Let $S$ be a smooth convex surface in $\mathbb{R}^3$
(although my question may as well be asked for the surface of a polyhedron).
Say that $\gamma$ is a *shortest halving curve* if
(a) it partitions the surface area of $S$ into two equal-area halves,
and (b) it is the shortest such curve (under the Euclidean metric).

Q. Is $\gamma$ necessarily a simple (non-self-intersecting) closed geodesic on $S$?

On a polyhedral surface, the equivalent would be a simple closed *quasigeodesic*
(in Alexandrov's sense).

**Addendum**. Douglas Zare's counterexample to that vain hope:

^{(Not metrically accurate.)}

The red circle seems to be a shortest halving curve, but it is not a geodesic.